Posterior Summary and Visualization Utilities

This vignette demonstrates the summary and plotting utilities available for stochtree models.

Setup

Load necessary packages

library(stochtree)

import numpy as np
import matplotlib.pyplot as plt
from stochtree import BARTModel, BCFModel, plot_parameter_trace

Set a seed for reproducibility

random_seed = 1234
set.seed(random_seed)

random_seed = 1234
rng = np.random.default_rng(random_seed)

Supervised Learning

We begin with the supervised learning use case served by the bart() function.

Below we simulate a simple regression dataset.

n <- 1000
p_x <- 10
p_w <- 1
X <- matrix(runif(n * p_x), ncol = p_x)
W <- matrix(runif(n * p_w), ncol = p_w)
f_XW <- (((0 <= X[, 10]) & (0.25 > X[, 10])) *
  (-7.5 * W[, 1]) +
  ((0.25 <= X[, 10]) & (0.5 > X[, 10])) * (-2.5 * W[, 1]) +
  ((0.5 <= X[, 10]) & (0.75 > X[, 10])) * (2.5 * W[, 1]) +
  ((0.75 <= X[, 10]) & (1 > X[, 10])) * (7.5 * W[, 1]))
noise_sd <- 1
y <- f_XW + rnorm(n, 0, 1) * noise_sd

n = 1000
p_x = 10
p_w = 1
X = rng.uniform(size=(n, p_x))
W = rng.uniform(size=(n, p_w))
# R uses X[,10] (1-indexed) = Python X[:,9]
f_XW = (
    ((X[:, 9] >= 0)    & (X[:, 9] < 0.25)) * (-7.5 * W[:, 0]) +
    ((X[:, 9] >= 0.25) & (X[:, 9] < 0.5))  * (-2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.5)  & (X[:, 9] < 0.75)) * ( 2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.75) & (X[:, 9] < 1.0))  * ( 7.5 * W[:, 0])
)
noise_sd = 1.0
y = f_XW + rng.standard_normal(n) * noise_sd

Now we fit a simple BART model to the data.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3
)
bart_model <- stochtree::bart(
  X_train = X,
  y_train = y,
  leaf_basis_train = W,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)

bart_model = BARTModel()
bart_model.sample(
    X_train=X,
    y_train=y,
    leaf_basis_train=W,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3
    },
)

We obtain a high level summary of the BART model by running print().

print(bart_model)

stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

print(bart_model)

BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function.

summary(bart_model)

stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.888, standard deviation = 0.049, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.7976738 0.8276973 0.8531417 0.8850262 0.9191620 0.9512392 0.9905685 
Summary of leaf scale posterior: 
3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
       2.5%         10%         25%         50%         75%         90% 
0.005152223 0.005820257 0.006539487 0.007328878 0.008172267 0.009171917 
      97.5% 
0.010802867 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = -0.064, standard deviation = 3.286, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-6.8485115 -4.9269938 -1.8268155 -0.1202225  2.0285138  4.2590778  6.5453792

print(bart_model.summary())

BART Model Summary:
-------------------
BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.934, standard deviation = 0.049, quantiles:
    2.5%: 0.845
   10.0%: 0.872
   25.0%: 0.901
   50.0%: 0.933
   75.0%: 0.966
   90.0%: 0.998
   97.5%: 1.037
Summary of leaf scale posterior: 3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
    2.5%: 0.005
   10.0%: 0.005
   25.0%: 0.006
   50.0%: 0.007
   75.0%: 0.007
   90.0%: 0.008
   97.5%: 0.009
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 0.106, standard deviation = 3.286, quantiles:
    2.5%: -6.737
   10.0%: -4.313
   25.0%: -1.950
   50.0%: 0.169
   75.0%: 2.093
   90.0%: 4.420
   97.5%: 6.575

None

We can use the plot() function to produce a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

plot(bart_model)

ax = plot_parameter_trace(bart_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function to pull the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), leaf scale \(\sigma^2_{\ell}\), in-sample mean function predictions y_hat_train) and then plot any subset or transformation of these values.

y_hat_train_samples <- extractParameter(bart_model, "y_hat_train")
obs_index <- 1
plot(
  y_hat_train_samples[obs_index, ],
  type = "l",
  main = paste0("In-Sample Predictions Traceplot, Observation ", obs_index),
  xlab = "Index",
  ylab = "Parameter Values"
)

y_hat_train_samples = bart_model.extract_parameter("y_hat_train")
obs_index = 0
fig, ax = plt.subplots()
ax.plot(y_hat_train_samples[obs_index, :])
ax.set_title(f"In-Sample Predictions Traceplot, Observation {obs_index}")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
plt.show()

Causal Inference

We now run the same demo for the causal inference use case served by the bcf() function in R and the BCFModel Python class.

Below we simulate a simple dataset for a causal inference problem with binary treatment and continuous outcome.

# Generate covariates and treatment
n <- 1000
p_X = 5
X = matrix(runif(n * p_X), ncol = p_X)
pi_X = 0.25 + 0.5 * X[, 1]
Z = rbinom(n, 1, pi_X)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[, 3]
tau_X = X[, 2] * 2 - 1

# Generate outcome
epsilon = rnorm(n, 0, 1)
y = mu_X + tau_X * Z + epsilon

# Generate covariates and treatment
n = 1000
p_X = 5
X = rng.uniform(size=(n, p_X))
pi_X = 0.25 + 0.5 * X[:, 0]
Z = rng.binomial(1, pi_X, n).astype(float)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[:, 2]
tau_X = X[:, 1] * 2 - 1

# Generate outcome
epsilon = rng.standard_normal(n)
y = mu_X + tau_X * Z + epsilon

Now we fit a simple BCF model to the data

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3, 
  adaptive_coding = TRUE
)
bcf_model <- stochtree::bcf(
  X_train = X,
  y_train = y,
  Z_train = Z,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)

bcf_model = BCFModel()
bcf_model.sample(
    X_train=X,
    Z_train=Z,
    y_train=y,
    propensity_train=pi_X,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3, 
      "adaptive_coding": True
    },
)

We obtain a high level summary of the BCF model by running print().

print(bcf_model)

stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

print(bcf_model)

BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function / method.

summary(bcf_model)

stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.949, standard deviation = 0.045, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.8643862 0.8907757 0.9182082 0.9484272 0.9783383 1.0078629 1.0409780 
Summary of prognostic forest leaf scale posterior: 
3000 samples, mean = 0.002, standard deviation = 0.000, quantiles:
        2.5%          10%          25%          50%          75%          90% 
0.0008568294 0.0010371220 0.0012421974 0.0014786088 0.0018064272 0.0021478841 
       97.5% 
0.0025004825 
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.437, mean (treated) = 1.029, standard deviation (control) = 0.348, standard deviation (treated) = 0.315
quantiles (control):
       2.5%         10%         25%         50%         75%         90% 
-1.17874376 -0.89652527 -0.66059102 -0.41437907 -0.18414640 -0.01349886 
      97.5% 
 0.16926139 
quantiles (treated):
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.4624619 0.6462934 0.8138740 1.0075352 1.2248498 1.4620051 1.6881949 
Summary of treatment effect intercept (tau_0) posterior: 
3000 samples, mean = 0.164, standard deviation = 0.734, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-1.3587447 -0.9591607 -0.2164569  0.1000003  0.7793058  1.1575356  1.3306564 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.487, standard deviation = 0.946, quantiles:
    2.5%      10%      25%      50%      75%      90%    97.5% 
1.654992 2.250739 2.792019 3.475277 4.131819 4.727274 5.379340 
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = 0.063, standard deviation = 0.524, quantiles:
       2.5%         10%         25%         50%         75%         90% 
-0.73251543 -0.55922780 -0.39695284 -0.03452641  0.57664883  0.79384361 
      97.5% 
 0.91214143

print(bcf_model.summary())

BCF Model Summary:
------------------
BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.872, standard deviation = 0.042, quantiles:
    2.5%: 0.794
   10.0%: 0.819
   25.0%: 0.843
   50.0%: 0.871
   75.0%: 0.900
   90.0%: 0.924
   97.5%: 0.958
Summary of prognostic forest leaf scale posterior: 3000 samples, mean = 0.002, standard deviation = 0.001, quantiles:
    2.5%: 0.001
   10.0%: 0.001
   25.0%: 0.001
   50.0%: 0.002
   75.0%: 0.002
   90.0%: 0.002
   97.5%: 0.003
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.575, mean (treated) = 1.114, standard deviation (control) = 0.365, standard deviation (treated) = 0.352
quantiles (control):
    2.5%: -1.346
   10.0%: -1.053
   25.0%: -0.801
   50.0%: -0.547
   75.0%: -0.324
   90.0%: -0.147
   97.5%: 0.093

quantiles (treated):
    2.5%: 0.501
   10.0%: 0.672
   25.0%: 0.875
   50.0%: 1.096
   75.0%: 1.330
   90.0%: 1.547
   97.5%: 1.868
Summary of treatment effect intercept (tau_0) posterior: 3000 samples, mean = 0.025, standard deviation = 0.584, quantiles:
    2.5%: -1.121
   10.0%: -0.909
   25.0%: -0.285
   50.0%: 0.087
   75.0%: 0.475
   90.0%: 0.737
   97.5%: 0.955
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.482, standard deviation = 0.961, quantiles:
    2.5%: 1.854
   10.0%: 2.210
   25.0%: 2.779
   50.0%: 3.475
   75.0%: 4.157
   90.0%: 4.775
   97.5%: 5.334
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = -0.028, standard deviation = 0.625, quantiles:
    2.5%: -1.071
   10.0%: -0.912
   25.0%: -0.587
   50.0%: 0.071
   75.0%: 0.540
   90.0%: 0.741
   97.5%: 0.920

None

In R, we have a plot() that produces a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

In Python, we provide a plot_parameter_trace() function for requesting a traceplot of a specific model parameter.

plot(bcf_model)

ax = plot_parameter_trace(bcf_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function in R or the extract_parameter() method in Python to query the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), prognostic forest leaf scale \(\sigma^2_{\mu}\), CATE forest leaf scale \(\sigma^2_{\tau}\), adaptive coding parameters \(b_0\) and \(b_1\) for binary treatment, in-sample mean function predictions y_hat_train, in-sample CATE function predictions tau_hat_train) and then plot any subset or transformation of these values.

adaptive_coding_samples <- extractParameter(bcf_model, "adaptive_coding")
plot(
  adaptive_coding_samples[1, ],
  type = "l",
  main = "Adaptive Coding Parameter Traceplot",
  xlab = "Index",
  ylab = "Parameter Values",
  ylim = range(adaptive_coding_samples),
  col = "blue"
)
lines(adaptive_coding_samples[2, ], col = "orange")
legend(
  "topright",
  legend = c("Control", "Treated"),
  lty = 1,
  col = c("blue", "orange")
)

adaptive_coding_samples = bcf_model.extract_parameter("adaptive_coding")
fig, ax = plt.subplots()
ax.plot(adaptive_coding_samples[0, :], color="blue", label="Control")
ax.plot(adaptive_coding_samples[1, :], color="orange", label="Treated")
ax.set_title("Adaptive Coding Parameter Traceplot")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
ax.legend(loc="upper right")
plt.show()